On the least norms of entire functions of exponential type with restricted initial conditions (Q1978835)

Let \(E_{p,\sigma }\), \(1\leq p\leq \infty \), \(\sigma >0\), be the set of all entire functions of exponential type which belong to \(L_p(-\infty ,\infty)\). The author presents an asymptotic estimate of the value \(\tau _{n,p,\sigma }=\sup \inf \|f\|_{L_p}\) as \(n\to \infty \), where the upper bound is taken over all complex-valued vectors \(a=(a_0,\dots ,a_n)\) with \(|a_s |\leq 1\), and the lower bound is taken over all \(f\in E_{p,\sigma }\) such that \(f^{(s)}(0)=a_s\), \(s\in \{0,\dots ,n\}\).